Worked examples
The question
Question: find the exact value of cos 150°.
1 Find the quadrant
We turn 150° from the positive x-axis, against the clock.
The ray meets the circle at P.
150° is between 90° and 180°.
So it is in quadrant II, the top left.
150° is in quadrant II
2 Find the reference angle
The reference angle is the angle between the ray and the x-axis.
Here it is 180° − 150°, which is 30°.
180° − 150° = 30°
Careful!
Always measure to the x-axis, never to the y-axis.
The y-axis gives 60°: the wrong triangle.
Careful: not 150° − 90° = 60° ✗
3 Find the sign
In quadrant II, P is left of the y-axis: x is negative.
P is above the x-axis: y is positive.
quadrant II : x −, y +
Cos is the across number x.
In quadrant II, x is negative.
So cos 150° = −cos 30°: the same size as for 30°, with a minus.
cos 150° = −cos 30°
4 Use the reference angle's value
From the unit circle: cos 30° = √3/2.
cos 30° = √32
So cos 150° = −√3/2.
cos 150° = −√32
5 Check
Let's check with the calculator: cos 150° is about −0.8660.
cos 150° ≈ −0.8660
And −√3/2 is about −0.8660.
The same!
So it is right.
−√32 ≈ −0.8660
Let's look back.
First: find the quadrant.
Then: find the reference angle.
Then: find the sign.
Then: use the reference angle's value.
Last: check.
▶ Watch it solved
The question
Question: find the exact value of sin 210°.
1 Find the quadrant
We turn 210° from the positive x-axis, against the clock.
The ray meets the circle at P.
210° is between 180° and 270°.
So it is in quadrant III, the bottom left.
210° is in quadrant III
2 Find the reference angle
The reference angle is the angle between the ray and the x-axis.
Here it is 210° − 180°, which is 30°.
210° − 180° = 30°
Careful!
Always measure to the x-axis, never to the y-axis.
The y-axis gives 60°: the wrong triangle.
Careful: not 270° − 210° = 60° ✗
3 Find the sign
In quadrant III, P is left of the y-axis: x is negative.
P is below the x-axis: y is negative.
quadrant III : x −, y −
Sin is the up number y.
In quadrant III, y is negative.
So sin 210° = −sin 30°: the same size as for 30°, with a minus.
sin 210° = −sin 30°
4 Use the reference angle's value
From the unit circle: sin 30° = 1/2.
sin 30° = 12
So sin 210° = −1/2.
sin 210° = −12
5 Check
Let's check with the calculator: sin 210° is about −0.5000.
sin 210° ≈ −0.5000
And −1/2 is about −0.5000.
The same!
So it is right.
−12 ≈ −0.5000
Let's look back.
First: find the quadrant.
Then: find the reference angle.
Then: find the sign.
Then: use the reference angle's value.
Last: check.
▶ Watch it solved
The question
Question: find the exact value of cos −60°.
1 Bring it into 0° to 360°
A negative angle turns the other way, with the clock.
Adding a full turn, 360°, lands at the same place: 300°.
−60° + 360° = 300°
Same place on the circle, same point.
So cos −60° = cos 300°.
cos −60° = cos 300°
2 Find the quadrant
We turn 300° from the positive x-axis, against the clock.
The ray meets the circle at P.
300° is between 270° and 360°.
So it is in quadrant IV, the bottom right.
300° is in quadrant IV
3 Find the reference angle
The reference angle is the angle between the ray and the x-axis.
Here it is 360° − 300°, which is 60°.
360° − 300° = 60°
Careful!
Always measure to the x-axis, never to the y-axis.
The y-axis gives 30°: the wrong triangle.
Careful: not 300° − 270° = 30° ✗
4 Find the sign
In quadrant IV, P is right of the y-axis: x is positive.
P is below the x-axis: y is negative.
quadrant IV : x +, y −
Cos is the across number x.
In quadrant IV, x is positive.
So cos 300° = cos 60°: the same size as for 60°, and plus.
cos 300° = cos 60°
5 Use the reference angle's value
From the unit circle: cos 60° = 1/2.
cos 60° = 12
So cos −60° = 1/2.
cos −60° = 12
6 Check
Let's check with the calculator: cos −60° is about 0.5000.
cos −60° ≈ 0.5000
And 1/2 is about 0.5000.
The same!
So it is right.
12 ≈ 0.5000
Let's look back.
First: bring it into 0° to 360°.
Then: find the quadrant.
Then: find the reference angle.
Then: find the sign.
Then: use the reference angle's value.
Last: check.
▶ Watch it solved
